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MATH 565X Supplementary Homework Spring 2013 12. Consider the underdetermined linear system Ax = b where A is an m × n matrix with m < n and b ∈ Rm . Assume the rows of A are linearly independent, so the rank of A equals the row rank of A equals m. There will then exist infinitely many solutions of the system, forming an n − m dimensional hyperplane in Rn . The minimum norm solution of Ax = b is the solution closest to the origin, which may be regarded as the solution of the constrained optimization problem min ||x||2 Ax=b a) Using the Lagrange multiplier method, derive the solution formula x = AT (AAT )−1 b. b) Find the minimum norm solution of the 3 × 5 system Ax = b when 1 2 0 3 2 4 A = −1 −1 4 2 0 b= 1 3 −2 2 1 1 −7 c) In Matlab, does x = A\b provide the minimum norm solution of Ax = b? If not, find out how to get it.